Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
522 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D251 |
| commutator, socle, cosocle | C23×C251 | C24 |
| Fitting | #396×C251 | C2 |
| hypercenter | #396 | D251 |
| hypocenter | C251 | #2825 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #396 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C251⋊#401 | C2 | semidirect |
| 2. #30631613 | C2 | semidirect |
| 3. #30631604 | C2 | semidirect |
| 4. #30631597 | C2 | semidirect |
| 5. #30631596 | C2 | semidirect |
| 6. #30631564 | C2 | semidirect |
| 7. #396×C251 | C2 | semidirect |
| 89. C251 | #2825 | semidirect |
| 93. #396 | D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C251⋊#401 | 2 |
| #30631613 | 2 |
| #30631604 | 2 |
| #30631597 | 2 |
| #30631596 | 2 |
| #30631564 | 2 |
| #396×C251 | 2 |
| #2825 | 251 |